# Continuous growth and decay

### Continuous growth and decay

We now have a better understanding of how the compounding frequency will affect the amount we wish to grow or decay. But what if we are dealing with something, say, that compounds every minute, second, or even millisecond? This concept is also known as continuous compounding. In this section, we will see a slight variation of an exponential growth and decay formula that models continuous exponential growth/decay.

#### Lessons

Continuous Growth/Decay: ${ A_f = A_i e^{rt}}$

${A_f}$: final amount

${A_i}$ : initial amount

${e }$
: constant = 2.718…

${r }$
: rate of growth/decay
• growth rate of 7% $\to {r = {7\over100} = 0.07}$
• growth rate of 15%$\to {r = - {15\over100} = - 0.15}$

${t }$
: total time given
• 1.
On Aiden's 10-year-old birthday, he deposited \$20 in a savings account that
offered an interest rate of 4% compounded continuously. How much money
will Aiden have in the account when he retires at the age of 60?

• 2.
A radioactive substance decays continuously. If the half-life of the substance
is 5 years, determine the rate of decay.